feat(init/data/list): filter theorems, non-meta qsort

library/init/data/list/lemmas.lean

@@ -629,6 +629,59 @@ by simp [scanr]
 
 attribute [simp] repeat take drop
 
+/- filter -/
+@[simp] theorem filter_nil (p : α → Prop) [h : decidable_pred p] : filter p [] = [] := rfl
+
+@[simp] theorem filter_cons_of_pos {p : α → Prop} [h : decidable_pred p] {a : α} :
+   ∀ l, p a → filter p (a::l) = a :: filter p l :=
+λ l pa, if_pos pa
+
+@[simp] theorem filter_cons_of_neg {p : α → Prop} [h : decidable_pred p] {a : α} :
+  ∀ l, ¬ p a → filter p (a::l) = filter p l :=
+λ l pa, if_neg pa
+
+@[simp] theorem filter_append {p : α → Prop} [h : decidable_pred p] :
+  ∀ (l₁ l₂ : list α), filter p (l₁++l₂) = filter p l₁ ++ filter p l₂
+| []      l₂ := rfl
+| (a::l₁) l₂ := if pa : p a then by simp [pa, filter_append] else by simp [pa, filter_append]
+
+@[simp] theorem filter_sublist {p : α → Prop} [h : decidable_pred p] : Π (l : list α), filter p l <+ l
+| []     := sublist.slnil
+| (a::l) := if pa : p a
+  then by simp[pa]; apply sublist.cons2; apply filter_sublist l
+  else by simp[pa]; apply sublist.cons; apply filter_sublist l
+
+@[simp] theorem filter_subset {p : α → Prop} [h : decidable_pred p] (l : list α) : filter p l ⊆ l :=
+subset_of_sublist $ filter_sublist l
+
+theorem of_mem_filter {p : α → Prop} [h : decidable_pred p] {a : α} : ∀ {l}, a ∈ filter p l → p a
+| []     ain := absurd ain (not_mem_nil a)
+| (b::l) ain :=
+  if pb : p b then
+    have a ∈ b :: filter p l, begin simp [pb] at ain, assumption end,
+    or.elim (eq_or_mem_of_mem_cons this)
+      (suppose a = b, begin rw -this at pb, exact pb end)
+      (suppose a ∈ filter p l, of_mem_filter this)
+  else
+    begin simp [pb] at ain, exact (of_mem_filter ain) end
+
+theorem mem_of_mem_filter {p : α → Prop} [h : decidable_pred p] {a : α}
+  {l} (h : a ∈ filter p l) : a ∈ l :=
+filter_subset l h
+
+theorem mem_filter_of_mem {p : α → Prop} [h : decidable_pred p] {a : α} :
+  ∀ {l}, a ∈ l → p a → a ∈ filter p l
+| []     ain pa := absurd ain (not_mem_nil a)
+| (b::l) ain pa :=
+  if pb : p b then
+    or.elim (eq_or_mem_of_mem_cons ain)
+      (suppose a = b, by simp [pb, this])
+      (suppose a ∈ l, begin simp [pb], exact (mem_cons_of_mem _ (mem_filter_of_mem this pa)) end)
+  else
+    or.elim (eq_or_mem_of_mem_cons ain)
+      (suppose a = b, begin simp [this] at pa, contradiction end) --absurd (this ▸ pa) pb)
+      (suppose a ∈ l, by simp [pa, pb, mem_filter_of_mem this])
+
 @[simp] lemma partition_eq_filter_filter (p : α → Prop) [decidable_pred p] : ∀ (l : list α), partition p l = (filter p l, filter (not ∘ p) l)
 | []     := rfl
 | (a::l) := by { by_cases p a with pa; simp [partition, filter, pa, partition_eq_filter_filter l],

library/init/data/list/qsort.lean

@@ -4,19 +4,42 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 -/
 prelude
-import init.data.list.basic
+import init.data.list.lemmas init.wf
 
 namespace list
 
-/- TODO(Leo): remove meta as soon as equation compiler has support for well founded recursion.
+-- Note: we can't use the equation compiler here because
+-- init.meta.well_founded_tactics uses this file
+def qsort.F {α} (lt : α → α → bool) : Π (x : list α),
+  (Π (y : list α), length y < length x → list α) → list α
+| []     IH := []
+| (h::t) IH := begin
+    ginduction partition (λ x, lt h x = tt) t with e large small,
+    have : length small < length (h::t) ∧ length large < length (h::t),
+    { rw partition_eq_filter_filter at e,
+      injection e,
+      subst large, subst small,
+      constructor;
+      exact nat.succ_le_succ (length_le_of_sublist (filter_sublist _)) },
+    exact IH small this.left ++ h :: IH large this.right
+  end
 
-   This is based on the minimalist Haskell "quicksort".
+/- This is based on the minimalist Haskell "quicksort".
 
    Remark: this is *not* really quicksort since it doesn't partition the elements in-place -/
-meta def qsort {α} (lt : α → α → bool) : list α → list α
-| []     := []
-| (h::t) :=
-  let (large, small) := partition (λ x, lt h x = tt) t
-  in qsort small ++ h :: qsort large
+def qsort {α} (lt : α → α → bool) : list α → list α :=
+well_founded.fix (inv_image.wf length nat.lt_wf) (qsort.F lt)
+
+@[simp] theorem qsort_nil {α} (lt : α → α → bool) : qsort lt [] = [] :=
+by rw [qsort, well_founded.fix_eq, qsort.F]
+
+@[simp] theorem qsort_cons {α} (lt : α → α → bool) (h t) : qsort lt (h::t) =
+  let (large, small) := partition (λ x, lt h x = tt) t in
+  qsort lt small ++ h :: qsort lt large :=
+begin
+  rw [qsort, well_founded.fix_eq, qsort.F],
+  ginduction partition (λ x, lt h x = tt) t with e large small,
+  simp [e], dsimp, rw [e]
+end
 
 end list